### Abstract

We consider the model of exploration of an undirected graph G by a single agent which is called the rotor-router mechanism or the Propp machine (among other names). Let π_{v} indicate the edge adjacent to a node v which the agent took on its last exit from v. The next time when the agent enters node v, first a "rotor" at node v advances pointer π_{v} to the edge which is next after the edge π_{v} in a fixed cyclic order of the edges adjacent to v. Then the agent is directed onto edge π_{v} to move to the next node. It was shown before that after initial O(mD) steps, the agent periodically follows one established Eulerian cycle, that is, in each period of 2m consecutive steps the agent traverses each edge exactly twice, once in each direction. The parameters m and D are the number of edges in G and the diameter of G. We investigate robustness of such exploration in presence of faults in the pointers π_{v} or dynamic changes in the graph. We show that after the exploration establishes an Eulerian cycle, (i) if at some step the values of k pointers π_{v} are arbitrarily changed, then a new Eulerian cycle is established within O(km) steps; (ii) if at some step k edges are added to the graph, then a new Eulerian cycle is established within O(km) steps; (iii) if at some step an edge is deleted from the graph, then a new Eulerian cycle is established within O(γm) steps, where γ is the smallest number of edges in a cycle in graph G containing the deleted edge. Our proofs are based on the relation between Eulerian cycles and spanning trees known as the "BEST" Theorem (after de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte).

Original language | English (US) |
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Title of host publication | Principles of Distributed Systems - 13th International Conference, OPODIS 2009, Proceedings |

Pages | 345-358 |

Number of pages | 14 |

DOIs | |

State | Published - Dec 31 2009 |

Event | 13th International Conference on Principles of Distributed Systems, OPODIS 2009 - Nimes, France Duration: Dec 15 2009 → Dec 18 2009 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 5923 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 13th International Conference on Principles of Distributed Systems, OPODIS 2009 |
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Country | France |

City | Nimes |

Period | 12/15/09 → 12/18/09 |

### Fingerprint

### Keywords

- Dynamic graphs
- Graph exploration
- Network faults
- Propp machine
- Rotor-router mechanism

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Principles of Distributed Systems - 13th International Conference, OPODIS 2009, Proceedings*(pp. 345-358). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5923 LNCS). https://doi.org/10.1007/978-3-642-10877-8_27